# Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

^{1}

^{2}

^{*}

## Abstract

**:**

Quod facile est in re, id probabile est in mente.—Gottfried Wilhelm Leibniz [1]

## 1. Consistency versus Non-Contradictoriness

## 2. Ci, a Logic of Formal Inconsistency

**Ci**is a member of the hierarchy of the LFIs with some features that make it reasonably close to classical logic; it is appropriate, in this way, to define a generalized notion of probability strong enough to enjoy useful properties. Consider the following stock of propositional axioms and rules:

**Definition**

**1.**

**Ci**(over Σ) is defined by the following Hilbert calculus:

**Axiom Schemas**

- $\mathbf{Ax}\mathbf{1}$
- $\alpha \to (\beta \to \alpha )$
- $\mathbf{Ax}\mathbf{2}$
- $(\alpha \to \beta )\to \left(\right(\alpha \to (\beta \to \gamma ))\to (\alpha \to \gamma \left)\right)$
- $\mathbf{Ax}\mathbf{3}$
- $\alpha \to (\beta \to (\alpha \wedge \beta \left)\right)$
- $\mathbf{Ax}\mathbf{4}$
- $(\alpha \wedge \beta )\to \alpha $
- $\mathbf{Ax}\mathbf{5}$
- $(\alpha \wedge \beta )\to \beta $
- $\mathbf{Ax}\mathbf{6}$
- $\alpha \to (\alpha \vee \beta )$
- $\mathbf{Ax}\mathbf{7}$
- $\beta \to (\alpha \vee \beta )$
- $\mathbf{Ax}\mathbf{8}$
- $(\alpha \to \gamma )\to \left(\right(\beta \to \gamma )\to ((\alpha \vee \beta )\to \gamma \left)\right)$
- $\mathbf{Ax}\mathbf{9}$
- $\alpha \vee (\alpha \to \beta )$
- $\mathbf{Ax}\mathbf{10}$
- $\alpha \vee \neg \alpha $
- $\mathbf{Ax}\mathbf{11}$
- $\circ \alpha \to (\alpha \to (\neg \alpha \to \beta \left)\right)$
- $\mathbf{Ax}\mathbf{12}$
- $\neg \neg \alpha \to \alpha $
- $\mathbf{Ax}\mathbf{13}$
- $\alpha \to \neg \neg \alpha $
- $\mathbf{Ax}\mathbf{14}$
- $\neg \circ \alpha \to (\alpha \wedge \neg \alpha )$

**Inference Rule**

**Modus Ponens (MP):**$\frac{\alpha ,\phantom{\rule{4pt}{0ex}}\alpha \to \beta}{\beta}$

**Ci**can be extended to the first-order logic

**QCi**(over a convenient extension of Σ) by adding appropriate axioms and rules.

**MP**define a Hilbert calculus for positive propositional classical logic (see [4]), and therefore, all of the laws concerning positive logic (as distribution of ∧ over ∨, etc.) are valid.

**Ci**, and $\alpha \equiv \beta $ means $\alpha {\u22a2}_{\mathbf{Ci}}\beta $ and $\beta {\u22a2}_{\mathbf{Ci}}\alpha $:

**Theorem**

**1.**

- 1.
- $\alpha \wedge (\beta \vee \gamma )\equiv (\alpha \wedge \beta )\vee (\alpha \wedge \gamma )$
- 2.
- $\alpha \vee (\beta \wedge \gamma )\equiv (\alpha \vee \beta )\wedge (\alpha \vee \gamma )$

**Proof.**

**Ci**cannot be semantically characterized by finite matrices, but it can be characterized in terms of valuations over $\{0,1\}$ (also called bivaluations):

**Definition**

**2.**

**Ci**. A function $v:\mathcal{L}\to \{0,1\}$ is a valuation for

**Ci**if it satisfies the following clauses:

**(Bival1)**- $v(\alpha \wedge \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\left(\beta \right)=1$
**(Bival2)**- $v(\alpha \vee \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\left(\beta \right)=1$
**(Bival3)**- $v(\alpha \to \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\left(\beta \right)=1$
**(Bival4)**- $v\left(\alpha \right)=0\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}v(\neg \alpha )=1$
**(Bival5)**- $v(\circ \alpha )=1\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v(\neg \alpha )=0$
**(Bival6)**- $v(\neg \neg \alpha )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=1$
**(Bival7)**- $v(\neg \circ \alpha )=1\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}v\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v(\neg \alpha )=1$

**Ci**is defined as expected: $\mathsf{\Gamma}\phantom{\rule{3.33333pt}{0ex}}{\vDash}_{\mathbf{Ci}}\phantom{\rule{3.33333pt}{0ex}}\phi $ iff, for every valuation v for

**Ci**, if $v\left(\gamma \right)=1$ for every $\gamma \in \mathsf{\Gamma}$, then $v\left(\phi \right)=1$.

**Theorem**

**2.**

**Proof.**

**Ci**as ${\sim}_{\beta}\alpha =\alpha \to {\perp}_{\beta}$, where ${\perp}_{\beta}=(\beta \wedge (\neg \beta \wedge \circ \beta ))$ is a bottom formula (that is: ${\perp}_{\beta}\text{}{\u22a2}_{\mathbf{Ci}}\psi $ for every ψ) for any sentence β. In order to simplify matters, a privileged β will be chosen, and the subscript will be omitted in ${\perp}_{\beta}$ and ${\sim}_{\beta}$ from now on.

**Theorem**

**3.**

**Ci**:

- 1.
- ${\u22a2}_{\mathbf{Ci}}\text{}\sim \alpha \to (\alpha \to \psi )$ for every α and ψ
- 2.
- ${\u22a2}_{\mathbf{Ci}}\alpha \text{}\vee \sim \alpha $
- 3.
- ${\u22a2}_{\mathbf{Ci}}\alpha \to \sim \sim \alpha $ and $\u22a2\sim \sim \alpha \to \alpha $
- 4.
- If $(\mathsf{\Gamma}{\u22a2}_{\mathbf{Ci}}\alpha \to \gamma )$ and $(\Delta {\u22a2}_{\mathbf{Ci}}\sim \alpha \to \gamma )$, then $(\mathsf{\Gamma},\Delta {\u22a2}_{\mathbf{Ci}}\gamma )$
- 5.
- ${\u22a2}_{\mathbf{Ci}}(\alpha \to \beta )\to (\sim \beta \to \sim \alpha )$ and so $\alpha \to \beta {\u22a2}_{\mathbf{Ci}}\text{}\sim \beta \to \sim \alpha $
- 6.
- ${\u22a2}_{\mathbf{Ci}}(\sim \alpha \to \sim \beta )\to (\beta \to \alpha )$, and so, $\sim \alpha \to \sim \beta \u22a2\beta \to \alpha $
- 7.
- $\u22a2(\alpha \to \sim \beta )\to (\beta \to \sim \alpha )$
- 8.
- $\u22a2\left(\right)open="("\; close=")">\phi \to (\alpha \to \beta )$
- 9.
- $\u22a2\left(\right)open="("\; close=")">\phi \to (\sim \alpha \to \sim \beta )$
- 10.
- $\u22a2\left(\right)open="("\; close=")">\phi \to (\alpha \to \sim \beta )$
- 11.
- $\sim \alpha \to \beta \u22a2\sim \beta \to \alpha $
- 12.
- $\u22a2\sim (\alpha \to \beta )\to (\alpha \wedge \sim \beta )$
- 13.
- $\u22a2\perp \to \alpha $

**Proof.**

**Ci**.

**Ci**hold as follows.

**Theorem**

**4.**

- 1.
- $\circ \alpha \u22a2\neg (\alpha \wedge \neg \alpha )$, but the converse does not hold
- 2.
- $\circ \alpha \u22a2\neg (\neg \alpha \wedge \alpha )$, but the converse does not hold
- 3.
- $\u22a2\circ \circ \alpha $
- 4.
- $\u22a2\circ \u2022\alpha $
- 5.
- $\u22a2\neg \u2022\circ \alpha $
- 6.
- $\u22a2\neg \u2022\u2022\alpha $
- 7.
- $\u22a2\circ \alpha \vee \alpha $
- 8.
- $\u22a2\circ \alpha \vee \neg \alpha $
- 9.
- $\u22a2\circ \alpha \vee \alpha \wedge \neg \alpha $
- 10.
- $\alpha \u22a2\alpha \wedge (\beta \vee \neg \beta )$
- 11.
- $\alpha \wedge (\beta \vee \neg \beta )\u22a2\alpha $

**Proof.**

**Theorem**

**5.**

**Ci**:

- 1.
- $\alpha \wedge \neg \alpha \wedge \circ \alpha \u22a2\beta $, for any β
- 2.
- $\circ \alpha \wedge \u2022\alpha \u22a2\beta $, for any β
- 3.
- $\circ \alpha \wedge \neg \circ \alpha \u22a2\beta $, for any β
- 4.
- $\u2022\alpha \wedge \neg \u2022\alpha \u22a2\beta $, for any β

**Proof.**

## 3. Consistency, Inconsistency and Paraconsistent Probability

**Definition**

**3.**

**L**, or a

**L**-probability function, is a function $P:\mathcal{L}\mapsto \mathbb{R}$ satisfying the following conditions, where ${\u22a2}_{L}$ stands for the syntactic derivability relation of

**L**:

- 1.
- Non-negativity: $0\le P\left(\phi \right)\le 1$ for all $\phi \in \mathcal{L}$
- 2.
- Tautologicity: If ${\u22a2}_{L}\phi $, then $P\left(\phi \right)=1$
- 3.
- Anti-tautologicity: If $\phi {\u22a2}_{L}$, then $P\left(\phi \right)=0$
- 4.
- Comparison: If $\psi {\u22a2}_{L}\phi $, then $P\left(\psi \right)\le P\left(\phi \right)$
- 5.
- Finite additivity: $P(\phi \vee \psi )=P\left(\phi \right)+P\left(\psi \right)-P(\phi \wedge \psi )$

**Theorem**

**6.**

**L**-probability measures.

- 1.
- If φ is a bottom particle in
**L**, then $P\left(\phi \right)=0$. - 2.
- If φ and ψ are logically equivalent in
**L**in the sense that $\phi \u22a2\psi $ and $\psi \u22a2\phi $, then $P\left(\psi \right)=P\left(\phi \right)$.

**Proof.**

**Theorem**

**7.**

**Ci**-probability measures.

**Ci**-probability measure; then:

- 1.
- $P(\alpha \vee \beta )=P\left(\alpha \right)+P\left(\beta \right)$, if α and β are logically incompatible.
- 2.
- $P(\circ \alpha )=2-\left(P\right(\alpha )+P(\neg \alpha \left)\right)$
- 3.
- $P(\alpha \wedge \neg \alpha )=P\left(\alpha \right)+P(\neg \alpha )-1$
- 4.
- $P(\sim \alpha )=1-P\left(\alpha \right)$
- 5.
- $P(\neg \circ \alpha )=1-P(\circ \alpha )$

**Proof.**

**Ci**and for the above defined notion of paraconsistent probability.

**Ci**.

**Ci**is (strongly) sound and complete with respect to such probabilistic semantics:

**Theorem**

**8.**

**Ci**with respect to probabilistic semantics:

**Proof.**

**Ci**proofs. For the right-to-left direction, notice that if $\mathsf{\Gamma}{\u22a9}_{P}\phi $, then in particular, this holds for the probability functions ${P}_{2}$, such that ${P}_{2}:L\mapsto \{0,1\}$. It suffices, then, by appealing to Theorem 2, to show that the mappings ${P}_{2}$ satisfy all of the conditions for bivaluations of Definition 2:

**(Bival1)**- ${P}_{2}(\alpha \wedge \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\left(\beta \right)=1$If ${P}_{2}(\alpha \wedge \beta )=1$, since ${P}_{2}(\alpha \wedge \beta )\le {P}_{2}\left(\alpha \right)$ and ${P}_{2}(\alpha \wedge \beta )\le {P}_{2}\left(\beta \right)$ by comparison (Definition 3) then ${P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\left(\beta \right)=1$.On the other hand, if ${P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\left(\beta \right)=1$, since again by comparison ${P}_{2}\left(\alpha \right)\le {P}_{2}(\alpha \vee \beta )$, then ${P}_{2}(\alpha \vee \beta )=1$, and the result follows by finite additivity, i.e., $P(\alpha \wedge \beta )=P\left(\alpha \right)+P\left(\beta \right)-P(\alpha \vee \beta )=1$.
**(Bival2)**- ${P}_{2}(\alpha \vee \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{or}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\left(\beta \right)=1$Analogous to the previous item, mutatis mutandis.
**(Bival3)**- ${P}_{2}(\alpha \to \beta )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{or}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\left(\beta \right)=1$By finite additivity ${P}_{2}(\alpha \vee (\alpha \to \beta ))+{P}_{2}(\alpha \wedge (\alpha \to \beta ))={P}_{2}\left(\alpha \right)+{P}_{2}(\alpha \to \beta )$. Since $\u22a2\phantom{\rule{3.33333pt}{0ex}}\alpha \vee (\alpha \to \beta )$, tautologicity implies ${P}_{2}(\alpha \vee (\alpha \to \beta ))=1$, and thus, as ${P}_{2}(\alpha \wedge (\alpha \to \phantom{\rule{3.33333pt}{0ex}}\beta ))\le {P}_{2}\left(\beta \right)$, ${P}_{2}\left(\beta \right)\ge {P}_{2}\left(\alpha \right)+{P}_{2}(\alpha \to \beta )-1$.Now, suppose ${P}_{2}(\alpha \to \beta )=1$; then, either ${P}_{2}\left(\alpha \right)=0$ or, if ${P}_{2}\left(\alpha \right)=1$, then ${P}_{2}\left(\beta \right)=1$.Reciprocally, suppose that either ${P}_{2}\left(\alpha \right)=0$ or ${P}_{2}\left(\beta \right)=1$. Since${P}_{2}(\alpha \vee (\alpha \to \beta ))+{P}_{2}(\alpha \wedge (\alpha \to \beta ))=1+{P}_{2}(\alpha \wedge (\alpha \to \beta ))={P}_{2}\left(\alpha \right)+{P}_{2}(\alpha \to \beta ),$ if ${P}_{2}\left(\alpha \right)=0$, then ${P}_{2}(\alpha \to \beta )\ne 0$, hence ${P}_{2}(\alpha \to \beta )=1$.If ${P}_{2}\left(\beta \right)=1$, since $\beta \u22a2(\alpha \to \beta )$, it follows immediately ${P}_{2}(\alpha \to \beta )=1$.
**(Bival4)**- ${P}_{2}\left(\alpha \right)=0\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}{P}_{2}(\neg \alpha )=1$Suppose that ${P}_{2}\left(\alpha \right)=0$. If ${P}_{2}(\neg \alpha )=0$, then by Theorem 7, Item (3), it follows that ${P}_{2}(\alpha \wedge \neg \alpha )=\phantom{\rule{3.33333pt}{0ex}}-1$, absurd.
**(Bival5)**- ${P}_{2}(\circ \alpha )=1\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{or}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}(\neg \alpha )=0$.Suppose that ${P}_{2}(\circ \alpha )=1$. If ${P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}(\neg \alpha )=1$, then Theorem 7, Item (2) leads to being absurd.
**(Bival6)**- ${P}_{2}(\neg \neg \alpha )=1\phantom{\rule{1.em}{0ex}}\u27fa\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=1$.This item follows directly from axioms $\mathbf{Ax}\mathbf{12}$ and $\mathbf{Ax}\mathbf{13}$ and tautologicity.
**(Bival7)**- ${P}_{2}(\neg \circ \alpha )=1\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}{P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}(\neg \alpha )=1$.Suppose that ${P}_{2}(\neg \circ \alpha )=1$. By Theorem 7, Item (5), ${P}_{2}(\circ \alpha )=0$. Hence, by the same Theorem 7, Item (2), ${P}_{2}(\circ \alpha )=0=2-({P}_{2}\left(\alpha \right)+{P}_{2}(\neg \alpha )$ and thus ${P}_{2}\left(\alpha \right)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{2}(\neg \alpha )=1$.

**Ci**is not an extension of classical propositional logic, just the contrary: although its language extends the language of classical logic, deductively, it is a contraction. Our (paraconsistent) probabilistic semantics to

**Ci**shows, therefore, that an alternative to truth-valued semantics in terms of non-standard probability functions can be provided even in cases of the contractions of classical logic.

## 4. Conditional Probabilities and Paraconsistent Updating

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Example**

**1.**

- D: the event that the drug test has declared “doping” (positive) for an individual;
- C: the event that the drug test has declared “clear” or “no doping” (negative) for an individual;
- A: the event that the person tested often uses the drug;
- $\neg A$: the event that the person tested does not often use the drug or has never used it.

If we view belief as generalized probability, then it makes sense to update beliefs but not combine them. If we view beliefs as a representation of evidence, then it makes sense to combine them, but not update them. This suggests that the rule of combination is appropriate only when we view beliefs as representations of evidence

## 5. Paraconsistent Probability Spaces

Countable additivity for probability has always been controversial. Émile Borel, who introduced it, and Andrei Kolmogorov, who confirmed its role in measure-theoretic probability, were both ambivalent about it. They saw no conceptual argument for requiring probabilities to be countably additive. It is merely mathematically convenient to assume they are. As Kolmogorov explained in his Grundbegriffe, countable additivity has no meaning for empirical experience, which is always finite, but it is mathematically useful. We can elaborate Kolmogorov’s explanation by pointing out that infinities enter into applied mathematics not as representation but as simplification.

**Definition**

**4.**

- 1.
- Ω is the sample space composed of all possible outcomes
- 2.
- $\Sigma \subseteq \wp (\Omega )$ is a set of events, such that Σ is a ${\sigma}^{\star}$-algebra, i.e.:
- (a)
- ∅ $\in \Sigma $, $\Omega \in \Sigma $ and $\Pi \in \Sigma $;
- (b)
- Σ is closed under ∪, ∩ and countable unions;
- (c)
- Σ is closed under the following two binary operations:
- i.
- ${X}^{\star}:\Sigma \mapsto \Sigma $, such that $\overline{X}\subseteq {X}^{\star}$, where $\overline{X}$ is the usual complement.
- ii.
- $\u25efX:\Sigma \mapsto \Sigma $, such that $\u25efX\cap {X}^{\star}=\overline{X}$

- (d)
- Π is the set of all consistent outcomes;

- 3.
- The map ${P}_{\mu}:\Sigma \mapsto [0,1]$ is a probability measure satisfying the following conditions:
- (a)
- ${P}_{\mu}(\Omega )=1$ and ${P}_{\mu}$ (∅) = 0
- (b)
- If ${S}_{1},\cdots {S}_{n}\in \Sigma $ are pairwise disjoint, then ${P}_{\mu}\left({\bigcup}_{i=1}^{\infty}{S}_{i}\right)={\sum}_{i=1}^{\infty}{P}_{\mu}\left({S}_{i}\right)$

Its success owes much to the mathematical convenience of making the calculus of probability merely a translation of modern measure theory. [...] No-one has given a real justification of countable additivity (other than just taking it as a ‘natural extension’ of finite additivity).

- $\left[\right[\phi \vee \psi \left]\right]=\left[\right[\phi \left]\right]\cup \left[\right[\psi \left]\right]$;
- $\left[\right[\phi \wedge \psi \left]\right]=\left[\right[\phi \left]\right]\cap \left[\right[\psi \left]\right]$;
- $\left[\right[\neg \phi \left]\right]=Val-\left[\right[\phi \left]\right]$.

**Example**

**2.**

- 1.
- Ω is any set (representing all possible outcomes)
- 2.
- $\Sigma \subseteq \wp (\Omega )$ is the set of all events, such that Σ is a σ-algebra, i.e.:
- (a)
- ∅ $\in \Sigma $, $\Omega \in \Sigma $ and $\Pi \in \Sigma $;
- (b)
- Σ is closed under ∪ and ∩;
- (c)
- Σ is closed under the binary operations:
- i.
- ${X}^{\star}=\overline{X}\cup \overline{\Pi}$
- ii.
- $\u25efX=\Pi $

**Ci**). When $\Pi =\phantom{\rule{3.33333pt}{0ex}}\Sigma $, this kind of paraconsistent probability space turns out to be a classical probability space. The structure $\langle \Omega ,\Sigma ,\Pi \rangle $ is a particular case of a paraconsistent algebra of sets, as investigated in [32], and the results therein (particularly Theorems 4 and 5) can be adapted to give a precise connection between the concepts of paraconsistent probability on events and probability on sentences in the

**Ci**logic (again, in the cases of finite additivity).

**Ci**). A deep investigation about paraconsistent probability spaces in the direction of a more sophisticated treatment involving random variables, measure theory and similar topics, however, is out of the scope of the present paper.

## 6. Discussion: From Paraconsistent Probability to Paraconsistent Possibility

## 7. Summary, Comments and Conclusions

**Ci**, taking profit from the underlying notion of consistency and essaying the first steps towards paraconsistent Bayesian updating.

**Ci**as well as paraconsistent probability theories based on several other LFIs. More recently, it has been shown in [42] that the Dutch book argument can be further extended to the domain of MV-algebras, providing a logical characterization of coherence for imprecise probability.

**Ci**enlarges the classical scenarios in significant ways: so for instance, even if impossible events should have degree zero by a rational agent, neither such events are necessarily unlikely nor a contradiction is an impossible event (although a consistent contradiction, as commented above, is impossible).

Studying axiomatizations of non-classical probabilities is an open-ended task. Can we extend the results of Paris, Mundici et al., and get more general sense of what set of axioms are sufficient to characterize expectations of truth value? A major obstacle here is the appeal throughout to the additivity principle (P3) and its variants, which is the only one to turn essentially on the behavior of particular connectives. Is there a way of capturing its content in terms of logical relations between sentences rather such hardwired constraints?

**LFI1**(see [44]): it is maximal with respect to classical logic (thus, in a certain sense closer to classical logic), enjoys some forms of De Morgan laws, and, above all, is algebraizable. This task is beyond the scope of this paper and will be postponed to future work.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Alarming Scenario | Cautious Scenario | Happy Scenario |

$P\left(A\right)=10\%$ | $P\left(A\right)=7.5\%$ | $P\left(A\right)=5\%$ |

$P(\neg A)=90\%$ | $P(\neg A)=92.5\%$ | $P(\neg A)=95\%$ |

$P(D/A)=98\%$ | $P(D/A)=98\%$ | $P(D/A)=98\%$ |

$P(D/\neg A)=10\%$ | $P(D/\neg A)=10\%$ | $P(D/\neg A)=10\%$ |

Result | Result | Result |

$P(A/D)=52\%$ | $P(A/D)=44\%$ | $P(A/D)=34\%$ |

Alarming Scenario | Cautious Scenario | Happy Scenario |

$P\left(A\right)=10\%$ | $P\left(A\right)=7.5\%$ | $P\left(A\right)=5\%$ |

$P(\neg A)=90\%$ | $P(\neg A)=92.5\%$ | $P(\neg A)=95\%$ |

$P(D/A)=98\%$ | $P(D/A)=98\%$ | $P(D/A)=98\%$ |

$P(D/\neg A)=2\%$ | $P(D/\neg A)=2\%$ | $P(D/\neg A)=2\%$ |

Result | Result | Result |

$P(A/D)=84\%$ | $P(A/D)=80\%$ | $P(A/D)=72\%$ |

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**MDPI and ACS Style**

Bueno-Soler, J.; Carnielli, W.
Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem. *Entropy* **2016**, *18*, 325.
https://doi.org/10.3390/e18090325

**AMA Style**

Bueno-Soler J, Carnielli W.
Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem. *Entropy*. 2016; 18(9):325.
https://doi.org/10.3390/e18090325

**Chicago/Turabian Style**

Bueno-Soler, Juliana, and Walter Carnielli.
2016. "Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem" *Entropy* 18, no. 9: 325.
https://doi.org/10.3390/e18090325